Temperature (K)
4.2 DFT Calculations
The DFT calculations were performed using the CASTEP [26] and the RPBE functional [28]. A Gaussian smearing width of σ = 0.1 eV was utilized in all calculations. The ion-electron interactions were approximated using ultrasoft pseudopotentials (usPP) with core corrections, calculations allowed for spin- polarized orbitals and a cut-off energy of 400 eV was set.
A five-layer slab with three layers relaxed was used with an optimized vacuum spacing of 12 Å between surfaces. K-point sampling was generated using the Monkhorst-Pack [29] procedure with a k-point spacing of <0.03 Å-1.
In order to determine if the converged energy was a minimum and not saddle point, a vibrational analysis was completed using a partial Hessian analysis [30] on the adsorbates in question. This is a valid approximation as the Fe atoms are significantly heavier than those of C, H and O. The atoms were perturbed by 0.005 Å in Cartesian space.
Adsorption energy was calculated as follows:
𝐸𝑎𝑑𝑠=𝐸(𝐹𝑒+𝑛𝑋)−𝐸(𝐹𝑒 𝑆𝑙𝑎𝑏)
𝑛 − 𝐸𝑋 (4.1)
Where E(Fe+nX) is the energy of n X adsorbates on an iron surface, E(Fe Slab) is the energy of a clean iron surface and EX is the energy of the adsorbate in the gas phase. It should be noted that for hydrogen the binding energy and adsorption energies are slightly difference since it involves dissociative adsorption.
Adsorption energy for hydrogen and oxygen was calculated as follows:
𝐸𝑎𝑑𝑠=𝐸(𝐹𝑒+𝑛𝑋)−𝐸(𝐹𝑒 𝑆𝑙𝑎𝑏)
𝑛 − 𝐸𝑋2 (4.2)
Where E(Fe+nH) is the energy of n X adsorbates on an iron surface, E(Fe Slab) is the energy of a clean iron surface and EH2 is the energy of X2 in the gas phase.
Binding energy for hydrogen and oxygen was calculated as follows:
𝐸𝑏𝑖𝑛𝑑 =𝐸(𝐹𝑒+𝑛𝑋)−𝐸(𝐹𝑒 𝑆𝑙𝑎𝑏)
𝑛 − 𝐸𝑋 (4.3)
Where E(Fe+nH) is the energy of n X adsorbates on an iron surface, E(Fe Slab) is the energy of a clean iron surface and EH is the energy of the X adsorbate in the gas phase.
The interaction between the adsorbates and the iron surface causes a shift in the iron atoms. The deformation of the iron surface was then calculated as follows:
𝐸𝑑𝑒𝑓𝑜𝑟𝑚= 𝐸[(𝐹𝑒+𝑛𝑋)−𝑛𝑋]− 𝐸(𝐹𝑒 𝑆𝑙𝑎𝑏) (4.4)
Where E[(Fe+nX)-nX] is the single point energy of the geometry optimized iron/adsorbate surface with adsorbates removed and E(Fe Slab) is the energy of a clean geometry optimized iron surface.
The results from this study were compared to those from Lo and Ziegler [6], Sorescu [7] and Govender [10,31]. It is important to note the difference in code and pseudopotential may yield different results. The accuracy of the different models can be seen by the calculation of bulk properties of Fe in Table 2-1. The computational methods used in these studies are summarized in Table 4-1.
The approximation of molecules in the gas phase can also give insight to the accuracy of the different models. In chapter 2, table 2-2 the bond angles, bond lengths and vibrational analysis of the molecules
70 calculated in a 10 Å x 10 Å x 10 Å box are compared with experimental gas phase molecules. We see that our results are accurate within 0.001 Å for bond lengths and 0.1o for bond length for CH, CH2 and CH4, while CH2 is accurate within 0.001 Å for bond lengths and 4 o for bond length. The same trend is seen for literature values.
71 Table 4-1: Details of computational methods used in studies by Lo and Ziegler[6], Govender[10,31], Bromfield et al. [9] and Sorescu[7].19
Lo and Ziegler [2] Govender et al. [6]/
Bromfield et al. [5] Sorescu [3]
Electron-ion interaction
Ultrasoft pseudo- potentials (usPP)
Projector-augmented
waves (PAW) PAW and USPP
Code VASP VASP VASP
Functional PW91 PW91 PW91 and PBE
Kinetic energy cut off 400 eV 400 eV 495 eV (usPP)
400 eV (PAW) Exchange correlation
energy GGA GGA GGA
k-point setting 7x7x1 for p(2x2) 5x5x1 for p(2x2) 4x4x2 for p(2x2)
Smearing width σ = 0.2 eV σ ≤ 0.1 eV σ = 0.1 eV
Vacuum thickness 10 Å 10 Å 10 Å
Spin polarized ✓ ✓ ✓
Slab approximations 5-layer slab (2 relaxed) 4-layer slab (1 relaxed) 7-layer slab (2 relaxed)
Vibrational analysis ✓ ✓ ✓
19 The model used in this study was as follows: USSP, CASTEP, RPBE, KE cut-off of 400 eV, GGA, 5x5x1 for p(2x2), σ ≤ 0.1 eV, vacuum thickness of 12 Å, spin polarized, 5-layer slab (2 relaxed), vibrational analysis was completed
72 4.2.1 Energetic breakdown
The energetic breakdown for the lateral interactions includes all the terms included in the Hamiltonian.
For CASTEP [26] the resulting energies are the kinetic energy, Hartree energy, local and non-local pseudopotential energies, exchange-correlation energy, Ewald energy and non-Coulombic energy.
Hartree, Ewald, pseudopotential and Non-Coulombic energies combined will give an overall electrostatic interaction. The exchange-correlation potential includes the effects of the Pauli Exclusion Principle and long range dipole interactions not described by the classical electrostatic interactions.
The kinetic energy gives us an idea of the shape of the electron density. A system with sharper changes in concavity will have a higher kinetic energy.
The energy contribution to the total electronic energy of each of the terms was considered and the change in each energy on adsorption was considered. This was calculated using the same procedure to calculate the integral adsorption energy was calculated. This means:
𝐸𝑖,𝑎𝑑𝑠 =𝐸𝑖,(𝐹𝑒+𝑛𝑋)−𝐸𝑖,(𝐹𝑒 𝑆𝑙𝑎𝑏)−𝑛∙𝐸𝑖,𝑋
𝑛 (4.5)
Where 𝑖 is either the kinetic energy, Hartree energy, local and non-local pseudopotential energies, exchange-correlation energy, Ewald energy or non-Coulombic energy.
4.2.2 The Fermi Energy
An important factor regarding the interactions of an electronic system is the position of the Fermi level of the system. The interactions of an adsorbates molecular orbitals with transition metal electron bands are dependent on the Fermi level. The classical molecular orbital theory can still be used to describe the interactions. Typically, we see that anti-bonding orbitals are higher in energy than bonding orbitals due to repulsions. In cases where metal bands are close to or intersected by the Fermi level, the anti-bonding orbitals can be higher than the Fermi level. Electrons can then be transferred to the metal and the repulsive forces diminished [23,34–36]. Van Steen and van Helden [23] showed that for CO and C on Fe (100) the Fermi level decreases with increasing coverage. Furthermore, the centre of the d-band is in the same position relative to the Fermi level, i.e. the overall energy of the d-band is lowered. As a result, the energy difference between the frontier orbitals (HOMO and LUMO) of the adsorbate changes. If these frontier orbitals are fully occupied, the Pauli repulsion would then increase.
4.2.3 Bader analysis
The Bader analysis was completed by reconstructing the algorithm proposed by Henkelman et al. [37] for Bader decomposition of charge density. The algorithm was reconstructed in Matlab and was built to receive the formatted density file (.den_fmt) which needs to be requested in the parameters file of the CASTEP setup.
73